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This paper presents a new lower bound for the discrete strategy improvementalgorithm for solving parity games due to Voege and Jurdziski. First, weinformally show which structures are difficult to solve for the algorithm.Second, we outline a family of games of quadratic size on which the algorithmrequires exponentially many strategy iterations, answering in the negative thelong-standing question whether this algorithm runs in polynomial time.Additionally we note that the same family of games can be used to prove asimilar result w.r.t. the strategy improvement variant by Schewe.

A Super-Polynomial Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it